For KK a number field, Artin’s reciprocity law says that this map is surjective, that it factors through the idele class groupK×\𝕀KK^\times \backslash \mathbb{I}_K and moreover that further quotienting this by the connected component 𝒪\mathcal{O} of 1 in the idele class group yields an isomorphism

∏′x∈ℂℂ((z−x))\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x)) (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)